Background: Let $ G $ be a subgroup of the orthogonal group $ O(n) $. If the action of $ G $ on the Lie algebra of $ O(n) $ is irreducible then either $ G $ is dense in $ SO(n) $ or $ G $ is finite. So if $ G $ is closed then $ G $ must be finite (or $ G=SO(n),O(n) $ but those cases are less interesting).
Anyway, suppose I have a finite subgroup $ G $ of $ O(n) $ and I want to determine if the action of $ G $ on the Lie algebra is irreducible. It is explained in Decomposition of $ V \otimes V^* $ for the natural representation that the adjoint representation of $ O(n) $ can be identified with the 2nd antisymmetric tensor power of the natural representation. So I think that we can test if the group $ G $ acts irreducibly on the Lie algebra by computing the 2nd exterior power of the natural character of $ G $ and checking to see if $ \Lambda^2(\chi) $ has norm 1.
However I tried this for some examples like the hyperoctahedral group $ 2 \wr S_4 $ in $ O(4) $ and it doesn't seem to be true. What am I doing wrong?
Or what is some easier/better way to check that a finite subgroup $ G $ of $ O(n) $ acts irreducibly in the adjoint representation?
